The article doesn’t say it’s going to be as big as the moon - it only says it’ll be 8x brighter. You’re also forgetting about the relative shapes of the objects: the moon, being round, reflects what light it does in all directions, while a designed mirror can be made to reflect it all in one specific direction.
All that said, I don’t think this will work, but not be of your math.
The relative sphericity of the moon is nearly immaterial here: when the moon is in full phase, do you see notable dimness around the edges where less sunlight is reflected away from you? Increasing the albedo to a near-perfect mirror cancels it all out.
Now, reflection won't be perfectly coliminated, but if you assume it is, then the mirror essentially becomes a piece of sun. Since the sun is about 400 000 times as bright as the moon, you'd need to scale the diameter of the sun by sqrt(400 000 / 8) =~ 223. Applying that scaling to your 5.4Km yields about 24m of diameter, which seems waaaay to small to me, so I am probably wrong.
That feels really
> about two Falcon Heavy trips
This does sound like it's within the state of the art! The mirror doesn't have to be one solid piece, it can be assembled from many smaller-ish satellites.
Of course you have overhead, so ... 5 Falcon Heavy trips? This is still very fine, if you're willing to put in the money.
Geosynch is 36,000 Km out.
The Moon's albedo is 0.12.
Let's assume we can get a mirror of albedo 0.96 -- this is a little better than polished silver. It's also conveniently 8 times as reflective as the moon.
The Moon's angular diameter is about 0.5 degrees. To match that angular diameter at 36,000 Km, we need a 5.4 Km wide object.
That doesn't sound like it's within the current state of the art. Maybe in another 10 years?
23 million square meters at 2 g/m^2 (aluminized mylar) is 46,000 Kg -- about two Falcon Heavy trips. That doesn't count any framework or booster or other infrastructure.